Complex Reaction Kinetics in Chemistry: A unified picture suggested by Mechanics in Physics Elena Agliari,1 Adriano Barra,2 Giulio Landolfi,3 Sara Murciano,4 and Sarah Perrone5 1Dipartimento di Matematica, Sapienza Universit`adi Roma, GNFM-INdAM Sezione di Roma, Italy 2Dipartimento di Matematica e Fisica Ennio De Giorgi, Universit`adel Salento, GNFM-INdAM Sezione di Roma, INFN Sezione di Lecce, Italy 3Dipartimento di Matematica e Fisica Ennio De Giorgi, Universit`adel Salento, INFN Sezione di Lecce, Italy 4Dipartimento di Matematica e Fisica Ennio De Giorgi, Universit`adel Salento, Italy, D´epartment de Physique, Ecole´ Normale Sup´eriore, France 5Dipartimento di Fisica, Universit`adi Torino, Italy (Dated: January 8, 2018) Complex biochemical pathways or regulatory enzyme kinetics can be reduced to chains of ele- mentary reactions, which can be described in terms of chemical kinetics. This discipline provides a set of tools for quantifying and understanding the dialogue between reactants, whose framing into a solid and consistent mathematical description is of pivotal importance in the growing field of biotechnology. Among the elementary reactions so far extensively investigated, we recall the so- called Michaelis-Menten scheme and the Hill positive-cooperative kinetics, which apply to molecular binding and are characterized by the absence and the presence, respectively, of cooperative interac- tions between binding sites, giving rise to qualitative different phenomenologies. However, there is evidence of reactions displaying a more complex, and by far less understood, pattern: these follow the positive-cooperative scenario at small substrate concentration, yet negative-cooperative effects emerge and get stronger as the substrate concentration is increased. In this paper we analyze the structural analogy between the mathematical backbone of (classical) reaction kinetics in Chemistry and that of (classical) mechanics in Physics: techniques and results from the latter shall be used to infer properties on the former. In particular, we first show that standard cooperative kinetics can be framed in terms of classical mechanics, and, interestingly, the emerging phenomenology, usually obtained by applying the thermodynamic principles, can be ob- tained by applying the principle of least action of classical mechanics. Further, since the saturation function (that is a bounded function) plays in Chemistry the same role played by velocity in Physics, we show how a relativistic scaffold naturally accounts also for the kinetics of the above-mentioned complex reactions. Of course, in the classical limit this generalized theory recovers the correct ana- lytical expressions of standard chemical kinetics. The reward in the proposed formalism is two-fold: a unique, consistent picture for cooperative-like reactions in chemical kinetics and a stronger and robust mathematical control, particularly useful to tackle reactions involving small numbers of molecules as those studied in nowadays experiments. PACS numbers: I. INTRODUCTION A. The Chemical Kinetics background The mathematical models that describe reaction kinetics provide chemists and chemical engineers with tools to arXiv:1801.01861v1 [q-bio.QM] 5 Jan 2018 better understand, depict and possibly control a broad range of chemical processes (see e.g., [27, 48]). These include applications to pharmacology, environmental pollution monitoring, food industry, etc. In particular, biological systems are often characterized by complex chemical pathways whose modeling is rather challenging and can not be recast in standard schemes [2, 3, 9, 10, 20, 23, 28, 36, 38, 42, 43, 55, 56] (see also [57, 59, 60, 64] for a different perspective). In general, one tries to split such sophisticated systems into a set of elementary constituents, in mutual interaction, and for which a clear formalization is available [4, 6, 39, 44, 58, 62]. In this context, one of the best consolidated, elementary scheme is given by the Michaelis-Menten law. This was originally introduced by Leonor Michaelis and Maud Menten to describe enzyme kinetics and can be applied to systems made of two reactants, say A (the binding molecule or, more generally, the binding sites of a molecule) and B (the free ligand, i.e., the substrate), which can bind (and unbind) to form the product AB. If we call S the concentration of free ligand, Y the saturation function (or fractional occupancy), namely the fraction of bound molecules (Y [0; 1]), and, accordingly, 1 Y the fraction of the unbound molecules, under proper assumptions, one can write 2 − S(1 Y ) = kY; (1) − 2 where k is the proportionality constant between response and occupancy (otherwise stated, it is the ratio between the dissociation and the association constants). In particular, as standard, it is assumed that the (a) the reaction is in a steady state, with the product being formed and consumed at the same rate, (b) the free ligand concentration is in large excess over that of the binding molecules in such a way that it can be considered as constant along the reaction, (c) all the binding molecules are equivalent and independent. Also, the derivation of the Michaelis-Menten law is based on the law of mass action. Reshuffling the previous equation we get Y = S=(S + k) which allows stating that k is the concentration of free ligand at which 50% of the binding sites are occupied (that is, when S = k, then Y = 1=2). Thus, denoting with S0 the half-saturation ligand concentration, we get S Y = : (2) S + S0 This equation represents a rectangular hyperbola with horizontal asymptote corresponding to full saturation, that is Y = 1; this is the typical outcome expected for systems where no interaction between binding sites is at work [63]. This model immediately settled down as the paradigm for Chemical Kinetics, somehow similarly to the perfect gas model (where atoms, or molecules - collisions apart - do not interact) of the Kinetic Theory in the early Statistical Physics [49]. Nevertheless, deviations from this behaviour were not late to arrive: the most common phenomenon was the occurrence of a positive cooperation among the binding sites of a multi-site molecule. Actually, many polymers and proteins exhibit cooperativity, meaning that the ligand binds in a non-independent way: if, upon a ligand binding, the probability of further binding (by other ligands) is enhanced, the system is said to show positive cooperativity. To fix ideas, let us make a practical example and let us consider the case of a well-known protein, i.e., the hemoglobin. This is responsible for oxygen transport throughout the body and it ultimately allows cellular respiration. Such features stem from hemoglobin's ability to bind (and to dislodge as needed) up to four molecules of oxygen in a non- independent way: if one of the four sites has captured an oxygen molecule, then the probability that the remaining three sites will capture further oxygen increases, and vice versa. As a result, if the protein is in an environment rich of oxygen (e.g., in the lungs), it readily binds up to four molecules of oxygen, and, as much readily, it gets rid of them when crossing an oxygen-deficient environment. To study quantitatively its behaviour one typically measures its characteristic input-output relation. This can be achieved by considering a set of M elementary experiments where these proteins, in the same amount for each experiment, are prepared in a baker and allowed to bind oxygen, which is supplied at different concentrations Si for different experiments (e.g., S1 < S2 < ::: < SM ). We can then construct a Cartesian plane, where on the abscissas we set the concentration of the ligand S (oxygen in this case, i.e. the input) while on the y-axes we put the fraction of protein bound sites Y (the saturation function, i.e., the output). In this way, for each experiment, once reached the chemical equilibrium, we get a saturation level and we can draw a point in the considered Cartesian plane; interpolating between all the points a sigmoidal curve will emerge (see Fig. 1). Archibald V. Hill formulated a description for the behavior of Y with respect to S: the so-called Hill equation empirically describes the fraction of molecules binding sites, occupied by the ligand, as a function of the ligand concentration [7, 8, 21, 65]. This equation generalizes the Michaelis-Menten law (2) and reads as kSnH Y = n ; (3) S0 + S H where nH is referred to as Hill coefficient and can be interpreted as the effective number of binding sites that are interacting each other. This number can be measured as the slope of the curve log[Y=(1 Y )] versus log(S), calculated − at the half-saturation point. Of course, if nH = 1 there is no cooperation at all and each binding site acts independently of the others (and, consistently, Michaelis-Menten kinetics is restored), viceversa, if nH > 1, the reaction is said to be cooperative (just like in hemoglobin), and if nH 1 the cooperation among binding sites is so strong that the sigmoid becomes close to a step function and the kinetics is named ultra-sensitive. The Michaelis-Menten law together with the extension by Hill, provided a good description for a bulk of chem- ical reactions, however, things were not perfect yet. For instance, some yeast's proteins (e.g., the Glyceraldehyde 3-Phosphate Dehydrogenase [24]) produced novel (mild) deviations from the Hill curve: for these enzymes, the coop- erativity of their binding sites decreases while increasing the ligand concentration. The following work by Daniel E. Koshland allowed understanding this kind of phenomenology by further enlarging the theoretical framework through the introduction of the concept of negative cooperativity. In fact, in the previous example, beyond the positive coop- eration between the binding sites there are also negative-cooperative effects underlying. Their effective action is to diminish the overall binding capabilities of the enzyme and thus to reduce the magnitude of its Hill coefficient.